OFFSET
0,2
FORMULA
E.g.f.: A(x) = 1/(1-2*x-x^2) * exp( Sum_{n>=1} -Pell(n-1)*x^n/n ).
E.g.f.: exp((1/sqrt(2))*arctanh(x*sqrt(2)/(1-x)))/sqrt(1-2*x-x^2). - Fabian Pereyra, Oct 19 2024
a(n) ~ 2^(3/2^(5/2) - 1/4) * sqrt(Pi) * (1 + sqrt(2))^(n + 1/2 - 1/2^(3/2)) * n^(n + 1/2^(3/2)) / (Gamma(1/2 + 1/2^(3/2)) * exp(n)). - Vaclav Kotesovec, Oct 28 2024
EXAMPLE
E.g.f.: A(x) = 1 + 2*x + 9*x^2/2! + 62*x^3/3! + 577*x^4/4! + 6762*x^5/5! +...
such that
log(A(x)) = 2*x + 5*x^2/2 + 12*x^3/3 + 29*x^4/4 + 70*x^5/5 + 169*x^6/6 + 408*x^7/7 + 985*x^8/8 +...+ A000129(n+1)*x^n/n +...
MATHEMATICA
nmax = 20; CoefficientList[Series[((1 - x + Sqrt[2]*x)/(1 - x - Sqrt[2]*x))^2^(-3/2) / Sqrt[1 - 2*x - x^2], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 28 2024 *)
PROG
(PARI) {Pell(n)=polcoeff(x/(1-2*x-x^2 +x*O(x^n)), n)}
{a(n)=n!*polcoeff(exp(sum(m=1, n, Pell(m+1)*x^m/m)+x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 13 2012
STATUS
approved